Date:         Tue, 30 Apr 1996 20:51:22 +0100
Reply-To:     John Whittington <johnw@MAG-NET.CO.UK>
Sender:       "SAS(r) Discussion" <SAS-L@UGA.CC.UGA.EDU>
From:         John Whittington <johnw@MAG-NET.CO.UK>
Subject:      Exponentiation of Negative Numbers

More mathematics than SAS, but .....
 
It seems as if SAS operates a blanket refusal to raise negative numbers to
non-integer powers, even in those situations (e.g. when the exponent is the
reciprocal of an odd integer) when it is a valid operation which should be
able to yield a real (i.e. non-'imaginery') solution, e.g.:
 
  x = (-32) ** 0.2 ;       which should evaluate to -2
 
This is a mild nuisance, but one can hard-code a workaround very easily.
 
In other situations (i.e. exponents which are NOT the reciprocals of odd
integers), I can certainly understand the problem.  Are there any
mathematicians out there who can tell me what, if anything, is the meaning
of a negative number raised to such a power?  With positive numbers, one can
conceptualise non-integral powers in terms of logarithms - but life is not
so easy when the numbers are negative;  one can, I suppose, try to think in
terms of the absolute value of the answer being the same as it would be for
the corresponding positive number, but what then would be its sign?  I
suspect that is far too simplistic, and that the answer is in fact a complex
number.
 
The point behind this is that it is clearly possible to have an algorithm
coded in SAS such that, with certain data, attempts would be made to raise a
negative number to a non integral power.  My question is - Is this *always*
going to represent an error condition - mathematically as well as in the
eyes of SAS?
 
In passing, for anyone who is not aware, I might remind that the syntax:
 
  x = (-32) ** 5 ;   is necessary because (unlike some other languages), SAS
interprets:
 
  x = -32 ** 5 ;  as meaning  x = - (32 ** 5) ;
 
 
John
 
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